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1
PRIME NUMBERS, TRIANGLES AND PARTICULAR RATIOS
7-11-13 * 11-13-17 * 41-53-67
PYTHAGORAS
Pythagoras many centuries after the pyramids will state the relation between the sides of the
particular right-angled triangle whose dimensions are respectively equal to 3,4 and 5, that is to
say three numbers separated by a constant interval, whose the square hypotenuse is equal to
the sum of the squares sides..
There is an infinity, i.e. all the triangles which are proportional to it such as 6,8,10, or 9,12,15,
etc... and of which the values of dimensions are separated by a constant interval.
The triangle known as of Pythagoras is present on several occasions in the constructions and
is not the result of the chance. . .
However Ancient Egypt don't know the multiplication nor division, consequently the squares
and the square roots. This knowledge for example could be issued from a systematic research
or be the consequence of the implementation of particular values.
To take an example, the peripheral rectangle which includes the 3 great pyramids is
distributed to the 3/5 by the Sphinx, which thus generates mathematically at least a triangle of
Pythagoras. (In reality it will be more interesting to consider what occurs starting from the
true grid).
What about 7 and 11?
The one who is for example a little interested in the pyramid of Cheops knows or could note
that its proportions result from the ratio 7/11. (Height 280 cubits either 7x40 and bases 440
cubits, or 11x40).
These are two prime numbers which are following and which generate a identical property to
the triangle of Pythagoras:7x7=49 plus 11x11=121 equal 170.
However the square root of 170 is 13,0384 for the hypotenuse.
i.e. with the hundredth precision a triangle 7-11-13 which is unique.
Does there exist others?
I am not informed of publication or unspecified evocation on this subject, and I met only three
mathematical continuations of prime numbers in this case when I had worked out a
calculation programme for my computer through the first 15.000 prime numbers and this with
continuations consisted of progressive intervals going from 1 to 500.
7-11-13 (hypotenuse: 13,0384)that is three prime numbers which are following, (4/100èmes
Error),
11-13-17 (hypotenuse: 17,0294) idem above, (3/100èmes Error),
2
41-53-67 (hypotenuse: 67,007) that is also three prime numbers which are following but with
an interval of 3, (7/1000èmes Error),
and there is no other: the simple fact of taking a multiple (triangle proportional) of these
values will lead that we will not use any more prime numbers, or they will not be integers.
2.700 years B.C. it could probably be only a research using the physical measurements
which could not have an absolute precision as our current calculators.
Only the three evoked triangles generate a continuation of the Pythagoras' type.
If one refers to the use of the 5 prime numbers, to a systematic research, and to the fact that
the multiplication and division were not known, a fortiori the square roots, this choice of 7/11
for Cheops could not be fortuitous.
But especially and it was an enormous surprise for me;
the Mathematical continuations
3,4,5 - 7,11,13 - 11,13,17 and 41,53,67
are not only physically present in the pyramids but they are
constituent elements from the outset in the design of Giza plateau!
Chephren's base lenght is 410,
the distance which separates the Eastern bases of Chephren and Mycerinus is 530
the distance which separates the Eastern bases of Chephren and Cheops is 670
The mathematical continuation 41-53-67 is dedicated to the whole Giza Plateau.
Continuing my research, it appeared to me that 7-11-13 was dedicated to Cheops each one
having seen like an obviousness the ratio 7/11 of the pyramid. (Height 280, Bases 440).
Its base can be also written 10 times 7-11-13 plus 13.
Concerning Chephren, it is less obvious and I have few elements about it except its
dimensions
It is probable that the mathematical continuation 11,13,17 is dedicated to Chephren;
One can simply note that its base is equal to 10 times the sum of 11-13-17.
These particular triangles of Pythagoras' type were well known and present just as I had
shown reports/ratios 4/5 of the perimeter of the three great pyramids subdivided into 2/3 by
the sphinx... (either generating triangles 3,4,5).
NUMBERS AND RATIOS
Proportions
3
They result from the use of the grids and a calculation system using the prime numbers.
I immediately point out that the values 41-53-67 are present at the basis of the concept,
and with the fact that this very particular mathematical continuation to my knowledge
had never made the object of any previous publication.
The layout which results and which I show on the following diagram are obviously only
partially illustrated.
Some of the values which result from the grids that I explained.
The Pyramids' location such as above shows obviously that it is not a question of a chance but
of a calculated system. And still for the example in the East-West direction some additional
relations:
4
The Mycerinus' eastern base is located at 10 times 41 cubits from the first enclosing Temenos'
wall; the distances separating the pyramids' eastern bases are respectively of 10 times 53
cubits for Mycerinus and Chephren and of 10 times 67cubits for Chephren and Cheops.
Cheops is located at 10 times 53 cubits from the Temple of the valley and as a remind,
Chephren has a base equal to 10 times 41 cubits.
As for the Chephren Western base which is located at 10 times 53 cubits from North-South
line passing by the Temenos.
A simple calculation will enable you to note other relations in the distances separating the
Western bases: 320 for Mycerinus-Chephren, then 640 (2 times 320) for Chephren-Cheops
and finally 960 (3 times 320) for Mycerinus-Cheops, i.e.1 time, 2 times and 3 times the
same value.
Others quite particular still exist and I show below only some partial examples of them:
From base
To base
Distance
Or
NorthCheops
South Chephren
1110
700+410
South Chephren
North Mycerinus
1110
260+410+440
South Cheops
South Chephren
670
260+410
East Cheops
East Chephren
670
230+440
5
East Chephren
East Mycerinus
530
410+120
The above components are among the most frequently met but are not restrictive.
Each one can check that these various values which govern a single reality are in
agreement with the whole system of 350 cubits grid subdivided in 11 times 30 cubits plus
one time 20. (Euclidean division).
All dimensions are rigorously in relations, it in the four directions, that confirms once more
that Mycerinus was quite present since the design!
Among the used Numbers:
Note: A prime number is a number which is divisible only by itself or by 1. Examples: 2, 3,
5,7,11
The values which we know are the pyramids' dimensions (400/280, 410/275 and 200/125) like
350 and 30 which represent the grid that I ha showed
If one seeks a relation between the values, a common denominator or something according to
our current methods, he will be very quickly stopped by the Chephren's base length which is
made up of 10 times 41 cubits, and 41 is a prime number which cannot be common or
component of the other pyramids.
The system seems abnormal, just like the fact that the pyramids despite their rigour and the
great pains taken to their realization are not aligned.
This system is deliberated. It was made so that the anomaly causes the question.
If the pyramids were aligned, there would be no interpellation and thus not research. I gave
the response with one of the grids. It is not they which are important, it is the system which is
not only deliberated but designed to transmit knowledge.
Idem for this value of 41.
If the question seems without answer it exists about it however at least one, but which
implies other methods, and the mathematicians will quickly recognize there the analogies with
"the algorithm of Euclid " and its positive and negative recursivity.
Perhaps its source of inspiration or a collected knowledge?
We used a grid of 350 cubits subdivided in 11 times 30 cubits plus one time 20, which can be
also written a grid of 2x5x5x7 subdivided in 11 times 2x3x5 and once 2x2x5.
If one wants to apply this same method for example to the components of dimensions of the
three large pyramids, the measurements could be written:
Base
Height
Cheops
(440)
2x5x5x7 plus 3 times 2x3x5
(280)
5x7 x2x2x2
Chephren
(410)
2x5x5x7 plus 2 times 2x3x5
(275)
5x5 x11
6
Mycerinus
(200)
2x5 x5x7 minus 5 times 2x3x5 (125)
5x5 x5
It is still a report.
This also has the merit of use only the 5 first of the prime numbers (2, 3, 5, 7, and 11) and to
be in coherence with the whole.
If one considers the architects' ruler of Ancient Egypt, one will notice that this ruler is mainly
divided into 28, in 7 and 2x2 as well as in simple ratios.
The constants and relations between the three large Pyramids could be written:
Constant 2x3x5
Secondary grid = 2x3x5 (30)
It appears as well in the bases as in the main grid.
Constant 5x5
Mycerinus' Height = 5 times 5x5 (125)
Chephren's Height = 11 times 5x5 (275)
Mycerinus' Base = 2x2x2 times 5x5 (200)
Constant 5x7
Cheops' Height = 2x2x2 times 5x7 (280)
Cheops' Base = 2x5 fois 5x7 plus 3 times2x3x5 (440)
Chephren's Base = 2x5 fois 5x7 plus 2 times2x3x5 (410)
Main grid = 2x5 times 5x7 (350)
One can also note that Chephren is the only one to use two constants (5x7 and 5x5) whereas
Cheops uses 5x7 and Mycerinus 5x5.
Chephren could be also written by using these two constants, which would give:
11x 5x5 =275 for its height,
11x 5x7 plus 5x5 = 410 for its base,
and also 2x2x2x2 times 5x5 (400) plus 2x5 for its base.
It would also seem that the first of the great pyramids (Cheops) was designed in terms of
dimensions with:
2x2x2x5x 11 for its base (440) and,
2x2x2x5x 7 for its Height (280).
7 and 11, remarkable numbers of which Pythagoras could not be unaware of!
The pyramid of Mycerinus would have been designed with:
7
5x5x 2x2x2
5x5x 5
One can also note that the height of each of the three Pyramids was designed with:
2x2x2x5x 7 for Cheops (280), it is also 10 times the 28 subdivisions of the Ancient Egypt
Architects' ruler.
5x5x 11 for Chephren (275)
5x5x 5 for Mycerinus (125)
A little exemple more:
Whereas the values of the two other pyramids require the use of the first five prime numbers,
Mycerinus requires only two of them: The first and the third. (Does not the ancient name of
the pyramid is “Mycerinus is Holy”?)
Its Height is written:
5x5x5 (Nowadays we would raised to a cubic power)
Its base is written: 2x2x2x5x5 (Nowadays we would say a cubic power by a square power)
A lot of relations whose complexity of the result has as a counterpoint only the simplicity of
the implementation...
There is still there no thesis; it is a simple measurable daily report, and many others one
exists.
Except the three mathematical continuations, only the first five prime numbers were used to describe the
entire whole.
8
A RELATION SUCH AS y=ax+b
The whole measurements result from a relation such as y=ax+b and we enumerate them in the
overall plan and the two tables below (or being the sum between them of various been worth
which appear in it).
Value
30
60
90
120
150
a
1
2
3
4
5
x
30
30
30
30
30
+b
0
0
0
0
0
Value
200
230
260
290
320
350
380
410
440
470
500
530
a
6
7
8
9
10
11
12
13
14
15
16
17
x
30
30
30
30
30
30
30
30
30
30
30
30
+b
20
20
20
20
20
20
20
20
20
20
20
20
Example: 670 can be consisted as 440 plus 230 or 410 plus 260, or 200 plus 210 plus 260, and so on. . .
9
IMPORTANT REMARK CONCERNING MEASUREMENTS
It is always difficult to obtain precise and reliable physical measurements. As I said, I used
only one copy of a Napoleon's chart and the apex angular statements published in the
literature.
If all the unit agrees, there is an exception which is the following one:
I do not have a sure knowledge of the distance which separates the parallels passing by the
Mycerinus'Eastern base and the Chephren Western base
If we use the angular statements of the Apex, this distance would equal to 120 cubits.
I met on Internet distances which can be more or less from 180 to 150 cubits but in these
cases there are several degrees of error in the statement of the two apex concerned.
Which would be the consequences if we took into account this value of 150 or 180 cubits ?
This would remain matching with the grids of 350 distributed according to the Euclidean
divison of 11 times 30 and with a complement of 20.
On the other hand, in this case we cannot any more speak about the perimetric rectangle
including the three pyramids in a ratio of 4/5 nor to say that mycerinus would have been
conceived in same time. However, the distribution in 3/5 and 2/5 in North to South direction
still remain.
It would result from it obviously a difference in the relation of the distances which connects
Mycerinus and the bases of the other pyramids, but this does not change anything with the
concept.
You can consult below these two cases..
Case of the distances (120 cubits) which matches with the angular statements...
10
Case of the distances such as 180 cubits then generating a ratio of 485/745, i.e: an angle of
33°3' 52 ("error " of 3°15' 19).
Below, what the Mycerinus’ relations become:
Case of the distances (120 cubits) which matches with the angular statements...
11
Case of the distances such as 180 cubits then generating a ratio of 485/745, i.e: an angle of
33°3' 52 ("error " of 3°15' 19).
On Line:
http://numerus.free.fr/ or http://gotogiza